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Question on Homotopical Structure on SimpSet

Posted by Urs Schreiber

More concretely, I’d like to get a better formal idea of the following situation:

in the standard model structure on SSetSSet, for every fibrant object X∈Kan⊂SSetX \in Kan \subset SSet (=∞\infty-groupoid) the object [Δ1,X][\Delta^1, X] is a path object for XX. So KanKan equipped with [Δ1,−][\Delta^1, -] is a category of fibrant objects in this sense with a functorial assignment of path objects.

In the Joyal model structure, for every fibrant object X∈WeakKan⊂SSetX \in WeakKan \subset SSet (= quasi-category) the object [Δ1,X][\Delta^1, X] clearly plays the role of the right directed path space object for XX, it still factors the diagonal as
X→[Δ1,X]→X×X,
X \to [\Delta^1, X] \to X \times X
\,,
but it is no longer a path object in the standard model-theoretic sense, as X→[Δ1,X]X \to [\Delta^1,X] need not be a weak equivalence anymore: Δ1\Delta^1 is a directed interval object.

I am thinking that there should be a good and nice relaxation of the axioms of category of fibrant objects such that WeakKanWeakKan becomes an example and such that the inclusion Kan↪WeakKanKan \hookrightarrow WeakKan becomes an inclusion of categories of fibrant objects in the relaxed sense.

There are some obvious guesses for how to try to relax the path space object axiom. But I am not entirely sure yet what the good way to do it really is. Has anyone thought about this?

Posted at March 5, 2009 8:13 PM UTC

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Re: Question on Homotopical Structure on SimpSet

Well, a different thing to do would be to replace Δ1\Delta^1 by the nerve N(J)N(J) of the walking isomorphism. Then you do get an interval object, in the model-category sense, for both model structures, so that Kan↪QCatKan \hookrightarrow QCat should then be an inclusion of categories of fibrant objects.

The groupoid {a→≃b}\{a \stackrel{\simeq}{\to} b\} (or its nerve) with invertible morphism is the standard un-directed interval of (quasi-)categories. You are saying one should use this to get a good notion of path objects.

Yes, I agree. But maybe my question was about something different:

I want to carry the path-space yoga which identifies homotopies (natural isomorphisms) between maps (functors) between ∞\infty-groupoids (Kan complexes) in terms of mapts into path objects over to the world of quasi-categories and have directed homotpies (i.e. natural transformations) encoded nicely in terms of maps into suitable directed path objects.

For this purpose N(a→≃b)N(a \stackrel{\simeq}{\to} b) is not the right path object. Maybe Δ1\Delta^1 isn’t either, though, but something directed like Δ1\Delta^1 should be needed.

Re: Question on Homotopical Structure on SimpSet

Good question. (As I said, my first comment was “a different thing to do.” (-: ) Here’s one possible approach. Consider, rather than merely a factorization of C→CI→C×CC\to C^I \to C\times C, a factorization

c•C→CΔ•→C•+1 c_\bullet C \to C^{\Delta^\bullet} \to C^{\bullet+1}

of simplicial objects in your category, where c•Cc_\bullet C is the constant simplicial object at CC and C•+1C^{\bullet+1} is the simplicial object whose nn-simplices are Cn+1C^{n+1}, with diagonals and projections giving the degeneracies and faces. Thus, we levelwise have factorizations

C→CΔn→Cn+1. C \to C^{\Delta^n} \to C^{n+1}.

When n=0n=0 we should probably demand that CΔ0=CC^{\Delta^0}=C. When n=1n=1 this looks like the path-object factorization. In general, the simplicial object CΔ•C^{\Delta^\bullet} is a candidate to be an “internal quasi-category” that encodes directed homotopies into CC and their compositions.

In any model category, there is a standard such factorization, namely the (acyclic-cofibration, fibration) factorization in the Reedy model structure on simplicial objects, and it is in fact an “internal Kan complex” in a suitable sense. I wouldn’t be much surprised if one could construct an analogous such factorization in any category of fibrant objects.

But now we care about cases when c•C→CΔ•c_\bullet C \to C^{\Delta^\bullet} is not a weak equivalence. It’s not clear to me whether in this case the whole simplicial object CΔ•C^{\Delta^\bullet} can be constructed from a factorization axiom for single maps. Also it seems less likely that the condition of CΔ•C^{\Delta^\bullet} being an “internal quasi-category” will come for free; we might have to assert it explicitly.

But it QCatQCat, at least, such a factorization does exist, where CΔnC^{\Delta^n} is what it looks like it should be, and it is an “internal quasi-category” in some sense. Probably I mean here that CΔn→CΛknC^{\Delta^n} \to C^{\Lambda^n_k} is a weak equivalence (or maybe an acyclic fibration) for each inner horn 0<k<n0\lt k\lt n, where CΛknC^{\Lambda^n_k} is constructed out of CΔ•C^{\Delta^\bullet} as a suitable limit.

Re: Question on Homotopical Structure on SimpSet

I don’t know, I wasn’t part of this debate. (-: Did it take place somewhere where I could have read it? I didn’t see it at “path object” or “interval object” on the nlab, but maybe I missed it.

Do you have some reason for thinking that CΔnC^{\Delta^n} should be anything different from the internal-hom [Δn,C][\Delta^n,C]? I don’t see any reason, at least not in QCatQCat, where it does give the right notion of composition of transformations.

Also, your condition on lifting modulo acyclic-fibration replacements doesn’t seem very different from requiring CΔn→CΛknC^{\Delta^n} \to C^{\Lambda^n_k} to be an acyclic fibration, since of course X^\hat{X} could just be a cofibrant replacement for XX, or even just the pullback of CΔnC^{\Delta^n} to XX. Are you just phrasing it that way so that you don’t have to talk about cofibrant objects? Saying that the map is an acyclic fibration seems a more natural condition to me.

Re: Question on Homotopical Structure on SimpSet

Are you just phrasing it that way so that you don’t have to talk about cofibrant objects?

Yes, I am finding it useful here to stay within the axiomnatics of categories of fibrant objects, not assuming anything about cofibrations.

I may be misled, but the cat of fib objects structure for instance on simplicial presheaves seems to be useful in that the fibrations are tractable (they are just localy Kan fibrations).

In the full model structure on simplicial presheaves the fibrant objects are the Kan-valued presheaves which satisfy descent, hence the fully ∞\infty-stackified things. This is much harder to handle in computations.

Re: Question on Homotopical Structure on SimpSet

I’m not an expert on simplicial presheaves, but I’m happy to try to work in the axiomatics of a category of fibrant objects. QCatQCat is, of course, also a good example of a category of fibrant objects in which the fibrations and weak equivalences are much simpler than they are in the larger model structure.

But I think that even within that axiomatics, asking for CΔn→CΛknC^{\Delta^n}\to C^{\Lambda^n_k} to be an acyclic fibration is reasonable. In particular, it implies the condition you suggested, since given X→CΛknX\to C^{\Lambda^n_k} you can take X^\hat{X} to be the pullback
X^→CΔn↓↓X→CΛkn.\array{\hat{X} & \to & C^{\Delta^n}\\
\downarrow && \downarrow\\
X& \to & C^{\Lambda^n_k}.}

Re: Question on Homotopical Structure on SimpSet

Hi Jim,

What I had in mind might be misguided, but whenever I think of directed spaces, I think of “spacetime”. The crucial aspects of spacetime can be encoded in a poset and a volume measure. I recently learned that directed spaces have a fundamental category.

With that in mind, I’m hoping to watch this drama play out so that one day, Urs or somebody will come out and say something like, “Ah ha! Spacetime IS a fundamental category!”

With that in mind, the picture I’m thinking of is a plane filled with either 2-cubes or 2-simplices. If we fill a plane with 2-simplices and then assign a direction to each 2-simplex as you suggested, i.e. 0→1→20\to 1\to 2, then many of these directions will wrap around on themselves and it is not clear that you could get a global sense of direction from these local directions that might represent the flow of time.

If you fill the plane with 2-cubes, then it is easy to see a global direction emerge, i.e. the direction across the diagonal of each 2-cube. Essentially, you have a directed binary tree (which Urs and I call a 2-diamond).

When I hear “directed space” I think “spacetime”. So although simplices have a local direction, it is not obvious to me that a global direction would emerge from a space filled with such directed simplices. I would interpret that global direction as a flow of time.

Like I said, maybe my thought was misguided, but I think that when you work with directed spaces, it might be helpful to think of a global direction as a flow (or direction) of time

with all the arrows between XXs acyclic fibrations, and with AIA^I a path object of AA and the maps on the right two boundary evaluation maps out of the path object. So this encodes that the map out of X^0\hat X_0 has a homotopty to the map out of X^1\hat X_1 after both are pulled back to a joint refinement of their domain.

And analogously for higher simplices.

First question: has this particular kind of construction of simplicial localization in cats of fib objects been considered before? Notice that using the morphisms given by the path object I can make a cosmetic modification to this diagram without changing its content

and it begins to look more akin to a hammock. It’s not hammock localization, but supposed to be something more direct using the properties of path objects in cats of fib objects.

Now, if CC here is something like locally Kan simplicial presheaves or presheaves with values in some other model of ∞\infty-groupoids, then C^(X,A)\hat C(X,A) will be a Kan complex, too.

But now assume CC is something like (presheaves with values in) weak Kan complexes. Then we are still in a category of fibrant objects, but now we are inclined to allow in the construction above AIA^Inot be a path object in the standard sense (not being weakly equivalent to AA), but be more generally of the form [D,A][D,A], where DD is some directed interval object. The point being that for objects in CC behaving (locally at least) like quasi-categories, we don’t want to demand the Hom SSetsSSetsC^(X,A)\hat C(X,A) to have outer fillers.

And there is another thing that makes me wonder, and triggers my question here:

if we drop the condition that AIA^I in the above hammock-like gadget be weakly equivalent to AA, then it seems natural to also drop in 1-cells the requirement that all the morphisms between the X^\hat Xs be acyclic.

without any condition on the left vertical morphisms is a bi-brane, namely a span of (generalized) spaces with a gerbe-like thing on each space and a morphism between the pullback of these to the correspondence space.

So I am wondering what’s going on. And if there is some general nonsense on directed path objects which would tell me which conditions to put here on the left vertical morphisms if on the right I allow directed path objects so that the whole thing somehow forms a nice structure.

Re: Question on Homotopical Structure on SimpSet

Regarding this particular kind of simplicial localization, it’s of course very much like the construction of simplicial mapping spaces in a model category using simplicial framings/resolutions. I haven’t seen it written down in this context.

I will continue to think about this, but one general comment: I do not expect that there will be a condition characterizing “directed path objects” solely in terms of the category-of-fibrant-objects structure. I’m not sure whether that’s what you’re expecting/hoping, but I wouldn’t expect it. My intuition is that categories-of-fibrant-objects, like model categories, are a presentation of (∞,1)(\infty,1)-categories, and if you want to recover a structure with noninvertible 2-cells from them, you need to impose extra structure, like considering an enriched or a closed-monoidal thingy (like QCatQCat).